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Can we find a sequence $u_n$ of positive real numbers such that $\sum_{n=1}^\infty u_n^2$ is finite, yet $\sum_{n=1}^\infty ({u_1+u_2+...+u_n\over n})^2$ is infinite ?

After several attempts, I think this is not possible, but I can't prove that the finiteness of the first sum implies the finiteness of the second sum.

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Hardy's inequality says $$\sum_{n=1}^{\infty}\left(\frac{a_1+\cdots+a_n}{n}\right)^p\le \left(\frac{p}{p-1}\right)^{p}\sum_{n=1}^{\infty}a_n^p$$ for any $p>1$.

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    $\begingroup$ You're welcome! A reference is, of course, Hardy-Littlewood-Polya's "Inequalities". $\endgroup$ – Gjergji Zaimi Jun 14 '10 at 23:37

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